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\newcommand{\C}{\mathbb{C}}
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\begin{document}
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\title{Introduction to Quantum Mechanics\\Lecture 1}
\author{Brian Greene\footnote{These lecture notes were TeX'd by Alexander Berenbeim and Carlos Hernandez}} % ADD YOUR NAME HERE IF YOU WORK ON THIS LECTURE's NOTES
\date{September 15, 2011}

\maketitle

\section{A Brief Review}
\subsection{Classical Mechanics Versus Quantum Mechanics}


In the first lecture, we reviewed physics in one dimensional space. In general, we will consider the classical mechanics case
\begin{equation}
\Newton
\end{equation} 
where $\vec{x}$ solves equations to produce trajectories, and the quantum mechanical case of
\begin{equation}
\label{schrodinger}
{i\hbar\frac{\partial\psi(\mathbf{x},t)}{\partial t}= \frac{-\hbar^2}{2m}\frac{\partial^2\psi(\mathbf{x},t)}{\partial \mathbf{x}^2}+V(\mathbf{x})\psi(\mathbf{x},0)}
\end{equation}
where $\psi(x,0)$ is the wave form at some initial time $t=0$. Furthermore, we determine the probability of a particle by the function
\begin{equation}
p(a,b)=\int\limits_a^b|\W|^2\,dx \mbox{ s.t. } \normalize
\end{equation}
\section{An Introduction to Quantum Evolution}

In classical mechanics, given a position function $\mathbf{x}(t)$ seen in Figure 1, we can look at the time evolution of a particle and retain both position and velocity. In Quantum Mechanics, as Schr\"odinger's equation evolves from $t_0$ to $t_{obs}$, we notice the wave function collapse from $t_{obs}-\epsilon$ when we observe the particle at $t_{obs}$, only to immediately reappear at $t_{obs+\epsilon}$ (Figures 2 to 5). The act of 'observation' has an apparently profound impact, and we don't even have an operational definition of observation! Let that sink in.
\begin{figure}[!h]\label{CM}
\centering
\includegraphics[width=0.25\textwidth]{deterministic.png}
\caption{In classical mechanics, we are able to determine a precise trajectory for an object given sufficient intial conditions.}
\end{figure}
\begin{figure}[!h]\label{S1}
\caption{ At $t_0$, the initial conditions for $\W$ are represented by the function $\psi(x,t_0)$ depicted above}
\end{figure}
\begin{figure}[!h]\label{S2}
\caption{ As the particle changes in time, and as $\epsilon\to0$, the wave function looks like $\psi(x,t_{obs-\epsilon})$...}
\end{figure}
\begin{figure}[!h]\label{C}
\caption{...only to collapse at $t_{obs}$ at position $x_{obs}(t_{obs})$.}
\end{figure}
\begin{figure}[!h]\label{S3}
\caption{And yet for $t_{obs+\epsilon}$ as $\epsilon \to 0$, the wave function $\psi(x,t_{obs+\epsilon})$ does not collapse.}
\end{figure}
Why does the wave function collapse? We don't know. And more perplexing, there is no means by which we can "draw the line" between Classical and Quantum mechanics. So even though the wave function collapses at the instant of observation, immediately after, $\W$ holds as seen in Figure 5. This is spooky, and counterintuitive, as the observation of a particle at $t_{obs}$ will never give you information about a particle's location immediately prior or post the act of observation. Moreover, Schr\"odinger's equation does not apply during the wave collapse of the observer.

This puts physicists in a conundrum, since we can no longer describe the electron's path as we can the path of a well understood projectile such as a baseball. Nor, does it seem, that we can hope to observe an electron's path at increasingly smaller intervals. For a sense of why this is hopeless, consider a measure theoretic argument.

For a time interval (0,1), with $m(0,1)=1$, consider each moment we observe the particle to be subtracted from the interval. So if we observe at $t=\frac{1}{2}$, we remove $\frac{1}{2}$ from (0,1). Now our interval is $(0,\frac{1}{2})\cup(\frac{1}{2},1)$, but it still measures to 1. Now suppose we could observe all $t\in\Q\cap(0,1)$. Now let us remove all of these observations  from the interval (0,1); we find that $m((0,1)-\Q\cap(0,1))=1$. Even if at each discrete moment, the wave function collapses, there are still uncountably many intervals where the wave function holds. This is to say that unless we have uncountably many time observations which completely cover an interval, there will still be intervals where the wave equation holds, and since $\W$ is a probability wave, the particle could be distributed anywhere in space between these observations, preventing us from ever observing the electron path.

Now all of this begs a few questions, namely:
\begin{itemize}
\item
Must the wave function be smooth?
\item
Need the wave function be continuous $\forall$ $x$ $\in \R$?
\item
Where does Schr\"odinger's equation come from?
\end{itemize}
For now, let's answer the last question. Ultimately, Eq.(\ref{schrodinger}) is confirmed by the data, and it was motivated by both the discovery of the photo-electric effect and the double slit experiment.
\section{$\hbar$ Explained}
\subsection{The Photo-Electric Effect}
The birth of Quantum Mechanics can be traced back to Einstein's seminal 1905 paper on the photo-electric effect. Up until this paper, many thought of light behaving solely as a wave and that the intensity of light striking a surface determined the energy of the electron, although  data did not confirm the mathematics. What Einstein found was that the \emph{frequency}, $\nu$, of the light determined the energy of the electrons emitted, while the \emph{intensity} of the light determined the number of electrons. That is, photons had both particle and wave behavior.
\begin{figure}[H]\label{Sym}
\centering
\includegraphics[width=0.5\textwidth]{pelec.jpg}
\caption{Photons of various frequencies hitting a metal sheet and, subsequently, causing electron emission. Note that the frequency of light determines the kinetic energy of the emitted electrons.}
\end{figure}
So, what explains this? Well, Einstein described light itself as being made up of particles called \emph{photons}. The energy of a photon was proportional to its frequency while the intensity of a light is the number of photons emitted per second. Following Max Planck's work on black-body radiation describing a minimum quanta of energy, $h$, required for action, Einstein was able to describe the photo-electric effect as 
\begin{equation}
E=h\nu
\end{equation}
By this definition of energy, it follows that the momentum of a photon can be described as
\begin{equation*}
 P=\cfrac{h}{2\pi}\cfrac{2\pi}{\lambda}
\end{equation*}
which implies that momentum is fundamentally related to wavelength.

Now we may introduce $\hbar$. Known as the reduced Planck constant, $\hbar$ is an experimentally verified universal constant that describes the size of energy quanta. It is defined as the following:
\begin{equation*}
\hbar \equiv \cfrac{h}{2\pi}
\end{equation*}
 For the purposes of this class, $\hbar \approx 1.054 \times 10^{-34} J\cdot s$, where $J$ and $s$ are just the everyday units of energy and time, respectively.

With $\hbar$ as the lastl piece of the puzzle, we can finally describe moment in an elegant fashion
\begin{equation}
 P=\hbar k
\end{equation}

\subsection{Quick Recap}

\begin{table}[h!]
\begin{center}
\begin{tabular}{| c | c | c |} 
\hline CM & Wave & Particle\\ 
\hline
$E=\frac{1}{2}mv^2$ & $\omega$ & E=$\hbar\omega$\\
$E=\cfrac{p^2}{2m}$ & $\lambda$ & P=$\cfrac{\nu}{\lambda}=\hbar k$  \\ 
\hline
\end{tabular}
\end{center}
\caption{Analogous terms between classical mechanics and quantum mechanics}
\end{table}



\subsection{The Double Slit Experiment}


Experimentally, you can measure the values of the wavelength $\lambda$. Consider Fig. 7, where 
\begin{equation*}
\begin{array}{c}
\sin\theta_n = \cfrac{\Delta \theta_n}{d} \Rightarrow \Delta \theta_n = d\sin\theta \text{,}\\ \\ \text{where } \Delta \theta_n =  \left\{\begin{array}{cc}n\lambda & \text{ intensity maxima}\\ (n+\frac{1}{2})\lambda & \text{ intensity minima} \end{array}\right. \text{, } n = 0, 1, 2, \dots
\end{array}
\end{equation*}
For $\Delta \theta=n\lambda$, where $\lambda$ is wavelength, we can note constructive and destructive wave interference by minding our $n$ labels, where $n$ denotes the brightest locations, and $n\pm\frac{1}{2}$ denotes the darkest locations. As a result, we can infer that for a large enough distance, $x$, from the slits the lateral distance to an interference maxima, $y_n$, can be written as
\begin{equation*}
y_n = \frac{n \lambda D}{d}
\end{equation*}
\begin{figure}[!h]
\centering
\includegraphics[width=0.5\textwidth]{dslit.jpg}
\caption{Double Slit Schematic}
\end{figure}
\begin{itemize}
\item Correlate this with the momentum of the electrons you're firing from the electron gun
\item That is, we can get an experimentally determined relationship between the wavelength of the quantum wave and the momentum of the particles.
\end{itemize}
This gives us the fundamental wave equation: $\cfrac{\partial^2 y}{\partial x^2}=\cfrac{1}{w}\cfrac{\partial^2y}{\partial t^2}$ where w $\equiv$ speed of the wave modulation.
\end{document}